If we take a general polynomial with complex coefficients:
$$C_n z^n+C_{n-1}z^{n-1}+\dots+C_1z+C_0$$
We can apply a general Möbius tranformation to the variable:
$$z=\frac{aw+b}{cw+d},~~~~a,b,c,d \in \mathbb{C}$$
This seems to be the most general transformation which preserves the degree of a polynomial equation (if we discount the degenerate cases, for example assuming $ad-bc=1$), since multiplying by $(cw+d)^n$ we obtain a polynomial in $w$:
$$A_n w^n+A_{n-1}w^{n-1}+\dots+A_1w+A_0$$
The first coefficient, if I'm not mistaken:
$$A_n=C_n a^n+C_{n-1}a^{n-1}c+C_{n-2}a^{n-2}c^2+\dots+C_1ac^{n-1}+C_0c^n$$
Is there a name for polynomials related by this transformation? Something in the form of "polynomial $P$ is ______ to $Q$ ", or "polynomials $P$ and $Q$ are ______ ".